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Properties

Label 1288.2.q.d
Level $1288$
Weight $2$
Character orbit 1288.q
Analytic conductor $10.285$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1288,2,Mod(737,1288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1288, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1288.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.q (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2847317803\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{3} + 3 q^{5} + q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{3} + 3 q^{5} + q^{7} - 11 q^{9} + 26 q^{13} + 7 q^{17} + 8 q^{19} - 7 q^{21} + 11 q^{23} - 6 q^{25} - 50 q^{27} - 6 q^{29} + 12 q^{31} - 2 q^{33} + 11 q^{35} + q^{37} + 21 q^{39} - 24 q^{41} + 18 q^{43} + 19 q^{45} + 17 q^{47} - 11 q^{49} - 19 q^{51} + 5 q^{53} - 42 q^{55} + 22 q^{57} + 33 q^{59} - 15 q^{61} + 13 q^{63} + 9 q^{65} + 5 q^{67} + 8 q^{69} - 18 q^{71} + 5 q^{73} + 44 q^{75} - 23 q^{77} - 11 q^{79} + 13 q^{81} - 102 q^{83} + 66 q^{85} + 4 q^{87} + 26 q^{89} - 20 q^{91} - 6 q^{93} + 19 q^{95} - 42 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1 0 −1.16480 − 2.01748i 0 −0.670343 + 1.16107i 0 −2.07324 − 1.64368i 0 −1.21350 + 2.10184i 0
737.2 0 −0.930231 − 1.61121i 0 1.87678 − 3.25068i 0 2.06001 + 1.66023i 0 −0.230659 + 0.399513i 0
737.3 0 −0.894556 − 1.54942i 0 −0.0785719 + 0.136091i 0 1.66436 − 2.05667i 0 −0.100461 + 0.174003i 0
737.4 0 −0.794794 − 1.37662i 0 0.518443 − 0.897970i 0 −0.634790 + 2.56847i 0 0.236604 − 0.409811i 0
737.5 0 −0.280072 − 0.485099i 0 −0.260600 + 0.451373i 0 2.58287 − 0.573374i 0 1.34312 − 2.32635i 0
737.6 0 0.355291 + 0.615383i 0 −0.912155 + 1.57990i 0 −2.37453 + 1.16688i 0 1.24754 − 2.16080i 0
737.7 0 0.795532 + 1.37790i 0 −1.41754 + 2.45525i 0 0.254013 − 2.63353i 0 0.234256 − 0.405744i 0
737.8 0 0.831319 + 1.43989i 0 1.67475 − 2.90075i 0 2.10483 − 1.60302i 0 0.117816 − 0.204064i 0
737.9 0 0.995432 + 1.72414i 0 0.788667 − 1.36601i 0 −0.634459 − 2.56855i 0 −0.481768 + 0.834447i 0
737.10 0 1.36726 + 2.36816i 0 −1.53730 + 2.66269i 0 0.159120 + 2.64096i 0 −2.23879 + 3.87770i 0
737.11 0 1.71962 + 2.97846i 0 1.51788 − 2.62904i 0 −2.60819 + 0.444214i 0 −4.41416 + 7.64555i 0
921.1 0 −1.16480 + 2.01748i 0 −0.670343 − 1.16107i 0 −2.07324 + 1.64368i 0 −1.21350 − 2.10184i 0
921.2 0 −0.930231 + 1.61121i 0 1.87678 + 3.25068i 0 2.06001 − 1.66023i 0 −0.230659 − 0.399513i 0
921.3 0 −0.894556 + 1.54942i 0 −0.0785719 − 0.136091i 0 1.66436 + 2.05667i 0 −0.100461 − 0.174003i 0
921.4 0 −0.794794 + 1.37662i 0 0.518443 + 0.897970i 0 −0.634790 − 2.56847i 0 0.236604 + 0.409811i 0
921.5 0 −0.280072 + 0.485099i 0 −0.260600 − 0.451373i 0 2.58287 + 0.573374i 0 1.34312 + 2.32635i 0
921.6 0 0.355291 − 0.615383i 0 −0.912155 − 1.57990i 0 −2.37453 − 1.16688i 0 1.24754 + 2.16080i 0
921.7 0 0.795532 − 1.37790i 0 −1.41754 − 2.45525i 0 0.254013 + 2.63353i 0 0.234256 + 0.405744i 0
921.8 0 0.831319 − 1.43989i 0 1.67475 + 2.90075i 0 2.10483 + 1.60302i 0 0.117816 + 0.204064i 0
921.9 0 0.995432 − 1.72414i 0 0.788667 + 1.36601i 0 −0.634459 + 2.56855i 0 −0.481768 − 0.834447i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.2.q.d ✓ 22
7.c even 3 1 inner 1288.2.q.d ✓ 22
7.c even 3 1 9016.2.a.bk 11
7.d odd 6 1 9016.2.a.br 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.q.d ✓ 22 1.a even 1 1 trivial
1288.2.q.d ✓ 22 7.c even 3 1 inner
9016.2.a.bk 11 7.c even 3 1
9016.2.a.br 11 7.d odd 6 1